I'm sure you can find more detailed info on the net, but generally speaking a brief gives information to a lawyer and a motion is given to a judge to request some legal action.
A brief is given to appellate judges by both sides of the dispute. Briefs do not necessarily ask for any particular legal action from the court. Instead, they are written by lawyers to educate judges on the relevant legal issues of a case. Briefs are written in a way that presents the legal issues persuasively in the hopes that the judge will interpret the law in a way favorable to your client.